Optimal. Leaf size=86 \[ \frac {a x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{5 c}-b d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )+b d x \sqrt {c+\frac {d}{x^2}}+\frac {1}{3} b x^3 \left (c+\frac {d}{x^2}\right )^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {451, 335, 277, 217, 206} \[ \frac {a x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{5 c}-b d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )+\frac {1}{3} b x^3 \left (c+\frac {d}{x^2}\right )^{3/2}+b d x \sqrt {c+\frac {d}{x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rule 277
Rule 335
Rule 451
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^4 \, dx &=\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^5}{5 c}+b \int \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx\\ &=\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^5}{5 c}-b \operatorname {Subst}\left (\int \frac {\left (c+d x^2\right )^{3/2}}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} b \left (c+\frac {d}{x^2}\right )^{3/2} x^3+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^5}{5 c}-(b d) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x^2}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=b d \sqrt {c+\frac {d}{x^2}} x+\frac {1}{3} b \left (c+\frac {d}{x^2}\right )^{3/2} x^3+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^5}{5 c}-\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )\\ &=b d \sqrt {c+\frac {d}{x^2}} x+\frac {1}{3} b \left (c+\frac {d}{x^2}\right )^{3/2} x^3+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^5}{5 c}-\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {1}{\sqrt {c+\frac {d}{x^2}} x}\right )\\ &=b d \sqrt {c+\frac {d}{x^2}} x+\frac {1}{3} b \left (c+\frac {d}{x^2}\right )^{3/2} x^3+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^5}{5 c}-b d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 81, normalized size = 0.94 \[ \frac {1}{15} x \sqrt {c+\frac {d}{x^2}} \left (\frac {3 a \left (c x^2+d\right )^2}{c}-\frac {15 b d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c x^2+d}}{\sqrt {d}}\right )}{\sqrt {c x^2+d}}+5 b \left (c x^2+4 d\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 203, normalized size = 2.36 \[ \left [\frac {15 \, b c d^{\frac {3}{2}} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (3 \, a c^{2} x^{5} + {\left (5 \, b c^{2} + 6 \, a c d\right )} x^{3} + {\left (20 \, b c d + 3 \, a d^{2}\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{30 \, c}, \frac {15 \, b c \sqrt {-d} d \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + {\left (3 \, a c^{2} x^{5} + {\left (5 \, b c^{2} + 6 \, a c d\right )} x^{3} + {\left (20 \, b c d + 3 \, a d^{2}\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{15 \, c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 140, normalized size = 1.63 \[ \frac {b d^{2} \arctan \left (\frac {\sqrt {c x^{2} + d}}{\sqrt {-d}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-d}} - \frac {{\left (15 \, b c d^{2} \arctan \left (\frac {\sqrt {d}}{\sqrt {-d}}\right ) + 20 \, b c \sqrt {-d} d^{\frac {3}{2}} + 3 \, a \sqrt {-d} d^{\frac {5}{2}}\right )} \mathrm {sgn}\relax (x)}{15 \, c \sqrt {-d}} + \frac {3 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a c^{4} \mathrm {sgn}\relax (x) + 5 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} b c^{5} \mathrm {sgn}\relax (x) + 15 \, \sqrt {c x^{2} + d} b c^{5} d \mathrm {sgn}\relax (x)}{15 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 99, normalized size = 1.15 \[ \frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (-15 b c \,d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )+15 \sqrt {c \,x^{2}+d}\, b c d +5 \left (c \,x^{2}+d \right )^{\frac {3}{2}} b c +3 \left (c \,x^{2}+d \right )^{\frac {5}{2}} a \right ) x^{3}}{15 \left (c \,x^{2}+d \right )^{\frac {3}{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 91, normalized size = 1.06 \[ \frac {a {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} x^{5}}{5 \, c} + \frac {1}{6} \, {\left (2 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{3} + 6 \, \sqrt {c + \frac {d}{x^{2}}} d x + 3 \, d^{\frac {3}{2}} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,\left (a+\frac {b}{x^2}\right )\,{\left (c+\frac {d}{x^2}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 5.25, size = 184, normalized size = 2.14 \[ \frac {a c \sqrt {d} x^{4} \sqrt {\frac {c x^{2}}{d} + 1}}{5} + \frac {2 a d^{\frac {3}{2}} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{5} + \frac {a d^{\frac {5}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{5 c} + \frac {b \sqrt {c} d x}{\sqrt {1 + \frac {d}{c x^{2}}}} + \frac {b c \sqrt {d} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{3} + \frac {b d^{\frac {3}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{3} - b d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )} + \frac {b d^{2}}{\sqrt {c} x \sqrt {1 + \frac {d}{c x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________